\chapter{Resonances}
\label{chap:Resonances}

\section{Zeldovich regularization}
\label{sec:zeldovich}

As an alternative to complex scaling, here we discuss Zeldovich regularization,
where the known asymptotic forms of wave functions in configuration
space are utilized to define a regularized inner-product.
Using this, we can implement CA-EC in configuration space, without the complications
of complex-scaled contours. As $\Ip(p)<0$ for resonance states,
the wave function is ever increasing and not square integrable.
They are said to belong in a ``Rigged Hilbert Space''. Inner products and
normalization factors of Gamow states can be calculated by inserting a regularization
factor inside integrals which are otherwise divergent \cite{delaMadrid:2008sr},
allowing us to work directly in the RHS.

Let,
%
\begin{equation}
    \label{eq:zeldovich}
    \braket{\phi_1}{\phi_2}
    =\lim_{\mu\to 0}\int_{0}^{\infty}e^{-\mu r^2}\phi_1(r)\phi_2(r)\,dr
\end{equation}
%
Now assume that after some $r>R$, the wave functions becomes approximately equal to
their asymptotic forms, $N_1 e^{ip_1r}$ and $N_2 e^{ip_2r}$ respectively. Then the
integral can be broken into two.
%
\begin{spliteq}
    \braket{\phi_1}{\phi_2} 
    & \approx \lim_{\mu\to 0}\int_{0}^{R}e^{-\mu r^2}\phi_1(r)\phi_2(r)\,dr \\
    & \quad +N_1N_2\lim_{\mu\to 0}
    \int_{R}^{\infty}e^{-\mu r^2}e^{ip_1r}e^{ip_2r}\,dr \\
\end{spliteq}
%
where $N_{1,2}$ can be determined from $N=\phi(r)/e^{ipr}$ for any $r>R$. Using
results from \cite{kukulin1989theory} for the second term, and taking the limit in
the first term, this simplifies as follows.
%
\begin{equation}
    \label{eq:easy}
    \braket{\phi_1}{\phi_2} \approx \int_{0}^{R}\phi_1(r)\phi_2(r)\,dr
    +\frac{\ii N_1 N_2 e^{\ii(p_1+p_2)R}}{p_1+p_2}
\end{equation}
%
If $\phi_1(r)$ and $\phi_2(r)$ are solutions to finite-ranged potentials with ranges
$a_1$ and $a_2$, this result can be made exact by taking $R=\max\{a_1,a_2\}$.
For other potentials, a sufficiently large $R$ has to be considered such that
$V(r)\approx 0$ for all $r>R$.

Now, we have all the necessary tools for performing Galerkin projection on to the
reduced basis, to arrive at a smaller generalized eigenvalue problem. Rest of
the procedure would continue similarly.

\subsection{RHA-EC for Zeldovich method}

RHA-EC is straightforward in this configuration space and we can directly construct
free wave functions of the form $e^{\ii p r}$ and include them in the basis.

\subsection{CA-EC for Zeldovich method}

\ny{I need to check this math again. For now, please ignore this section.}

There will be no saving of additional memory usage by implementing CA-EC in
un-complex-scaled bases. Since RHA-EC already works better than CA-EC and is much
easier to implement, there is no reason to go with CA-EC,
other than for sanity checks, for which this section is dedicated.

\begin{lemma}
    For bound states, complex conjugation is equivalent to analytically rotating
    the function by an angle $2\phi$.
\end{lemma}

Consider a bound state training point $\ket{\psi}$.
%
\begin{equation}
    \ket{\psi}=\int_{0}^{\infty} dp\,p^2\, u(p)\ket{p}
\end{equation}
%
Now complex-scale the integration contour by an angle $-\phi$ (i.e., $\phi$ in the
clockwise direction).
%
\begin{equation}
    \ket{\psi}=\int_{-\phi} dp\,p^2\, u(p)\ket{p}
\end{equation}
%
This $u(p)$ is the reduced radial wave function that we obtain when we solve for
eigenstates with complex scaling. Now, consider the complex-conjugated wave function
on this contour.
%
\begin{equation}
    \ket{\Tilde{\psi}}=\int_{-\phi} dp\,p^2\, u^*(p)\ket{p}
\end{equation}
%
Since $\ket{\psi}$ is a bound state, $u(p)$ is real on the positive real axis.
Therefore, Schwarz reflection principle can be invoked to say $u^*(p)=u(p^*)$.
%
\begin{spliteq}
    \ket{\Tilde{\psi}}&=\int_{-\phi} dp\,p^2\, u(p^*)\ket{p}\\
    &=\int_{-\phi} dp\,p^2\, u(pe^{2i\phi})\ket{p}
\end{spliteq}
%
Since $u(pe^{2\ii\phi})$ is an analytic function, we can complex-scale the
contour back to the positive real axis.
%
\begin{equation}
    \ket{\Tilde{\psi}}=\int_{0}^{\infty} dp\,p^2\, u(pe^{2\ii\phi})\ket{p}
\end{equation}
%
The end result $\ket{\Tilde{\psi}}$ is simply the original state $\ket{\psi}$
rotated by an angle $2\phi$ on the complex $p$-plane.

Therefore, the equivalent of complex-conjugated vectors for the Zeldovich method
can be obtained by solving the Schr\"odinger's equation along a complex-scaled contour
($re^{-2\ii\phi}$) for some arbitrary angle $\phi$. As mentioned above, this method
is very inefficient in the Zeldovich implementation compared to RHA-EC.